THE OSCULATING SPACES OF A
CERTAIN CURVE IN [n]
by W. L. EDGE
(Received 27th February 1973)
1
The curve in question is the non-singular intersection T of the n — \ quadric
primals
£
akjXj = 0, k = 0, 1,2, ..., n - 2 ,
(1.1)
j = 0
where it is presumed that no two of the n+1 numbers a,- are equal. Define
f^)
= {<j>-ao){^> -«,)•••(<£-«„);
then it will be seen that the osculating prime of T at x = £, is
0.
(1.2)
Indeed, equations will be given for all the osculating spaces [s], such a space
being determined by n—s linearly independent linear equations. But (1.2)
is mentioned at the outset because equations for the osculating plane in [3]
and for the osculating solid in [4] are already known. The equation of the
osculating plane of an elliptic quartic curve in [3] is given by Salmon (3, p. 380);
the coefficients f'(a)t;3 appear there on taking a' = b' = c' = d' = 1. The
equation of the osculating solid of a special canonical curve in [4] is given
by Edge (2, p. 278), whose more prolix equation is seen, for n = 4, to be
equivalent to (1.2) here on substituting, from what is there labelled (2.2), in
the equation written as l,(p+ajg)2^jXj = 0. It was the belated perception of
this that suggested (1.2), and there is no difficulty in an a posteriori verification.
This relies on two circumstances.
;
(1) If ak = Za)lf'(aj)
then
(T0 = (T1 = ... = < 7 , , _ t = 0 .
(1.3)
This is proved by using the partial fractions for <j)k/f(<t>).
(2) The equations (1.1) may be regarded as n— 1 linear equations for the
/i + l " unknowns " x); they are, no two ai being equal, linearly independent
and so have n+l — (n— 1) = 2 linearly independent solutions. Clearly, in
virtue of (1.3), two such solutions are
x? = l / / ' ( a , ) and x) = ajlf'(aj).
Hence, whatever number 9 may be, other than the n + l critical values — a,,
the n + l equations
j
(1.4)
40
W. L. EDGE
give, by the alternative signing of n+1 square roots, a batch of 2" points on T.
One such batch has 6 = oo. If, differentiation being imminent, one scruples
to treat this batch as on a par with others there is the alternative use of
>
(1.4')
when the batch corresponds to (f> = 0.
2
Equation (1.2) is established if it can be shown that, for/? = 0, 1, ..., n— 1,
this means that, for all these n values of p,
H0+ajr-2Sjd>Zj = 0.
(2.1)
For/? = 0 this is so, by (1.1). Otherwise one repeatedly differentiates
this determines, for each./, one of the two analytic branches of (0+a,)* according
to the square root chosen and, having made the choice, one adheres thereto in
the subsequent differentiations. Then, on this understanding,
with Ap a non-zero constant, so that
and, so long as p ^ n — 1, (2.1) holds because of (1.3). This establishes the
validity of (1.2), at least for finite values of 6. But one can also differentiate
the square roots of the two sides of (1.4') and so arrive at
d'tjy/f'iaj)
and
from which the desired conclusion follows.
from glossing the text by references to (1.4').
Henceforward we may refrain
3
The same reasoning, however, applies to the equation
i:{f'{a$'-H)r-*d% = 0,
(3.1)
or
for any r such that 0 < r ^ n. Since this last relation is an identity in 9 for
p = 0, 1, ..., r—1 the points
£,, d£, d2^, ..., ds£
all satisfy those 5 equations (3.1) for which r = n, n—\, ..., 5+1. These
THE OSCULATING SPACES OF A CERTAIN CURVE IN [n] 41
therefore, with Xj replacing dp£j, are the n —s equations determining the osculating [s] of r at £. That they do determine the [s] is consequent on their linear
independence; that they are linearly independent follows once (4.3) below,
and the proceedings relating to it, have been noted.
4
One can now calculate Rs, the sth rank of F, i.e. the number of spaces
[j] that osculate F and meet a given [«—5— 1]. If this [n~s—1] is determined
by the s+1 linear equations
ai,o*o + a ;,i*i + -••+a.-,n^/i = 0 (' = n-s + l, ..., n + l)
(4.1)
it is met by those [s] which osculate F at points whose coordinates £ cause a
certain (« + l)-rowed determinant A to be zero: column j+1 of A consists of
{/'(a,.)}-2^"-3, {/'(a,)}"- 3 ^- 5 , .... { / W ' S r
1
(4-2)
followed by a n - s +i, ;••«„+1, y. The n—s numbers (4.2) all have the factor
2 1
residual factors are, in virtue of (1.4),
) n - s - \ (0 + a ; ) - 5 - 2 , ..., 9 + aj, 1.
If these are now multiplied in order by
(4.3)
and the products added, the sum is
One next performs a similar operation that does not involve the leading
member in (4.2); omit the leader in (4.3) and multiply the others in order
by
i-s-2
and add the products; the sum is a"~s~2. And so on. The whole procedure
transforms A, without changing its value, into a determinant having, so long
as i ^ n—s, in row i and column j+1 the element
The remaining s+l rows are still filled, as originally, by the coefficients of
(4.1). It now appears, by Laplace expansion on these s+1 rows, that the degree
of A in the coordinates £,- is (n—s)(2s— 1).
5
When E, is replaced by x, A = 0 becomes the equation of a primal whose
2"~1(n—s)(2s— 1) intersections with F are those points at which the osculating
|Y| intersects the [n—s—Y] given by (4.1). And so
Rs = 2"-\n-s)(2s-l).
42
W. L. EDGE
In particular: the class of F, or the number of its osculating primes passing
through an arbitrary point is
a classical result for n = 3 (there are 12 osculating planes of an elliptic quartic
through an arbitrary point in [3]) and obtained for n — 4 in (2). Also: the
order of the primal generated by the osculating [n—2]'s of F is
Rn.2=2\2n-5),
of course classical for n = 3 (the tangents of an elliptic quartic generate a
scroll of order 8). For n = 4 it follows that the osculating planes of the canonical
model of Humbert's plane sextic generate a threefold of order 48. There will
be, for each n, a single equation for Rn-2, presumably obtainable by some
process of elimination.
6
There is a more sophisticated procedure for determining the ranks Rs^
and perhaps it should be described. In order to apply it one must know the
genus n of F and a certain formula for the number of points, in the sets of a
linear series g on F, of multiplicity exceeding the freedom r of g; and indeed
a precise rule for calculating the number of times a multiple point of specified
singularity has to be counted.
As for the genus of F it is known (5, p. 83) that the canonical series of grade
2n — 2 is cut, on the complete non-singular intersection of genus n of n— 1
primals in [n], by primals of order nl+n2 + ... + nn-1 — (« + l), where the «,are the orders of the primals through the curve. Since, for F, each nx is 2 the
canonical series is cut by primals of order n — 3 and so
as stated by Baker (1, p. 185).
Take now any [n—s— 1]; the primes through it cut on F a linear series g
of grade 2"" 1 and freedom s. If a prime contains an osculating [s] of F, then
the contact with F counts s+l times in the corresponding set of g. The
standard formula (4, p. 85) for the number of points of multiplicity s+l in
a linear series of grade 2"" 1 and freedom s on a curve of genus n is
( S + 1){2-1 + ^ - 1 ) 5 } ;
for F this is
2"-2(s + l){2+(n-3)s}.
(6.1)
7
This, however, is not Rs because, whatever [«—s— 1] is chosen, there are
certain points W on F where osculating spaces of dimension less than s have
(s+l)-point intersection; these spaces can be joined to points in [n—s— 1]
THE OSCULATING SPACES OF A CERTAIN CURVE IN [n] 43
by spaces of dimension less than n. One has to know two things: how many
points W there are, and how much each contributes to the number (6.1).
Identification of these W is easy; they are the intersections of T with the
«+1 bounding primes x = 0 of the simplex of reference. Once this has been
proved it follows that there are 2"~1(n + l) of them, so that if each contributes
m to (6.1)
£ s = 2"- 2 [(s+l){2 + («-3)s}-2m(n + l)].
(7.1)
First, then, to note the special attributes of the points W.
F is its own harmonic inverse in each vertex X and opposite bounding
prime x = 0 of the simplex of reference: if P is on T, then XP is a chord of T
since it contains the image P' of P in the inversion. So the tangent of T at any
point W contains a vertex X. But the osculating plane co at W is the limiting
position of the plane joining this tangent to a neighbouring point ? of T;
since this plane contains both X and P it contains P'; co has 4-point intersection
with F at W. Similar reasoning shows the osculating solid to have 6-point
intersection, and so on, the osculating \s~\ having 2^-point intersection.
Now let B be an [n —s— 1]. The join [n—s] of B to any W lies in oo5"1
primes; of these, some, to be accounted for in a moment, are special, but the
"general" prime among these 00 s " 1 has only a single intersection with T
at W. However, the 00s~2 primes containing B and the tangent of T at Wall
have 2-point intersection, the 00s ~3 primes containing B and the osculating
plane of T at W all have 4-point intersection, and so on, until one has the single
prime, spanned by B and the osculating [s— 1] of T at W, having (25—2)-point
intersection. Then the rule, due to Corrado Segre (4, p. 86; for a textbook
reference see 6, p. 131) prescribes that, in such circumstances, W contributes
m = 1+2+4 + ...+2(s-l)-±s(s+l) = $(s-l)(s-2)
to the number (6.1). When %(s— l)(s—2) is substituted for m in (7.1) one finds,
as obtained by more elementary methods earlier,
Rs = 2"-\n-s)0.s-V).
REFERENCES
(1) H. F. BAKER, Principles of Geometry, Vol. 4 (Cambridge, 1925 and 1940).
(2) W. L. EDGE, The osculating solid of a certain curve in [4], Proc. Edinburgh
Math. Soc. (2) 17 (1971), 277-280.
(3) G. SALMON, A Treatise on the Analytic Geometry of Three Dimensions (Dublin,
1914).
(4) C. SEGRE, Introduzione alia geometria sopra un ente algebrico semplicemente
infinite Annali di Matematica (2) 22 (1894), 41-142; Opere I (Rome, 1957), 198-304.
44
W. L. EDGE
(5) F. SEVERI, S U alcune questioni di postulazione, Rend. Circ. Mat. Palermo,
17 (1903), 73-103.
(6) F. SEVERI, Trattato di geometria algebrica (Bologna, 1926).
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